Geodesic
A sharp inequality for the order of the minimal positive harmonic function in $T$-homogeneous domain
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004), pp. 375-379.

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Let $G$ be a simply connected domain in $\mathbb C$ which is $T$-homoheneous, i.e., $TG=G$ for some $T>0$. Let $\rho(G)$ be the order of the minimal positive harmonic function in $G$. We prove that a kind of symmetrization of $G$ and prove that it does not increase $\rho(G)$. This implies a sharp lower bound for $\rho(G)$ in terms of conformal modulus of a quadrilateral naturally connected with $G$. 
@article{JMAG_2004_11_a0,
     author = {V. Azarin and A. Gol'dberg},
     title = {A sharp inequality for the order of the minimal positive harmonic function in $T$-homogeneous domain},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {375--379},
     publisher = {mathdoc},
     volume = {11},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2004_11_a0/}
}
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V. Azarin; A. Gol'dberg. A sharp inequality for the order of the minimal positive harmonic function in $T$-homogeneous domain. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 11 (2004), pp. 375-379. http://geodesic.mathdoc.fr/item/JMAG_2004_11_a0/